Solve frac{sin(70^{circ})}{cos(70^circ)}cos(10^circ)left(sqrt{3}sin(20^circ)/cos(20^circ)-1right)

$$\frac{ \sin ( 70 ^ { \circ } ) }{ \cos ( 70 ^ { \circ } ) } \cos ( 10 ^ { \circ } ) \left( \sqrt{ 3 } \frac{ \sin ( 20 ^ { \circ } ) }{ \cos ( 20 ^ { \circ } ) } -1 \right)$$

Evaluate

$$\frac{1978194442373402692684302306858552108254022523\sqrt{3}}{2008711280270435822081894034612000000000000000}-\frac{578385361498952009063502667829}{213762589578543000000000000000}\approx -1$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{0.9396926207859083}{0.3420201433256688} 0.984807753012208 {(\sqrt{3} \frac{0.3420201433256687}{0.9396926207859084} - 1)}$$

Expand \frac{0.9396926207859083}{0.3420201433256688} by multiplying both numerator and the denominator by 10000000000000000.

$$0.984807753012208\times \frac{9396926207859083}{3420201433256688}\left(\sqrt{3}\times \frac{0.3420201433256687}{0.9396926207859084}-1\right)$$

Convert decimal number 0.984807753012208 to fraction \frac{61550484563263}{10000000000}. Reduce the fraction \frac{61550484563263}{10000000000} to lowest terms by extracting and canceling out 1.

$$\frac{61550484563263}{62500000000000}\times \frac{9396926207859083}{3420201433256688}\left(\sqrt{3}\times \frac{0.3420201433256687}{0.9396926207859084}-1\right)$$

Multiply \frac{61550484563263}{62500000000000} times \frac{9396926207859083}{3420201433256688} by multiplying numerator times numerator and denominator times denominator.

$$\frac{61550484563263\times 9396926207859083}{62500000000000\times 3420201433256688}\left(\sqrt{3}\times \frac{0.3420201433256687}{0.9396926207859084}-1\right)$$

Do the multiplications in the fraction \frac{61550484563263\times 9396926207859083}{62500000000000\times 3420201433256688}.

$$\frac{578385361498952009063502667829}{213762589578543000000000000000}\left(\sqrt{3}\times \frac{0.3420201433256687}{0.9396926207859084}-1\right)$$

Expand \frac{0.3420201433256687}{0.9396926207859084} by multiplying both numerator and the denominator by 10000000000000000.

$$\frac{578385361498952009063502667829}{213762589578543000000000000000}\left(\sqrt{3}\times \frac{3420201433256687}{9396926207859084}-1\right)$$

Use the distributive property to multiply \frac{578385361498952009063502667829}{213762589578543000000000000000} by \sqrt{3}\times \frac{3420201433256687}{9396926207859084}-1.

$$\frac{578385361498952009063502667829}{213762589578543000000000000000}\sqrt{3}\times \frac{3420201433256687}{9396926207859084}+\frac{578385361498952009063502667829}{213762589578543000000000000000}\left(-1\right)$$

Multiply \frac{578385361498952009063502667829}{213762589578543000000000000000} times \frac{3420201433256687}{9396926207859084} by multiplying numerator times numerator and denominator times denominator.

$$\frac{578385361498952009063502667829\times 3420201433256687}{213762589578543000000000000000\times 9396926207859084}\sqrt{3}+\frac{578385361498952009063502667829}{213762589578543000000000000000}\left(-1\right)$$

Do the multiplications in the fraction \frac{578385361498952009063502667829\times 3420201433256687}{213762589578543000000000000000\times 9396926207859084}.

$$\frac{1978194442373402692684302306858552108254022523}{2008711280270435822081894034612000000000000000}\sqrt{3}+\frac{578385361498952009063502667829}{213762589578543000000000000000}\left(-1\right)$$

Multiply \frac{578385361498952009063502667829}{213762589578543000000000000000} and -1 to get -\frac{578385361498952009063502667829}{213762589578543000000000000000}.

$$\frac{1978194442373402692684302306858552108254022523}{2008711280270435822081894034612000000000000000}\sqrt{3}-\frac{578385361498952009063502667829}{213762589578543000000000000000}$$